New failure mechanism for evaluating ultimate inclined load adjacent to slope

A new failure mechanism is proposed for calculating the ultimate inclined load adjacent to the slope, i.e., the slope is in the limit state when the critical slope contour and the slope surface are at the critical position where two intersections will occur. The conventional view is that the critical slope contour calculated by the method of characteristics has only a concave shape. This study found that the critical slope contour changes from concave to convex when the inclined load imposed on the slope top surface increases. The feasibility of the proposed method is verified by the finite element limit analysis (FELA) and the definition of the ultimate load. The parametric analysis showed that the current method of characteristics (CMOC) overestimated the ultimate inclined load and gave an incorrect conclusion since it assumed larger failure models at a low strength ratio or large friction angle. The proposed method does not require assumption or search of the failure models, and it can solve the shortcomings of CMOC.


Introduction
Many structures, e.g., buildings, bridge abutments, and transmission line towers, are built near slopes, and the foundations of the structures are usually subjected to the inclined load. It is a complex problem to calculate the ultimate inclined load of a shallow foundation adjacent to the slope. The inclined load and slope reduce the bearing capacity of the soil [1]. Current research gives exact solutions or empirical equations for the effect of inclined loads on the bearing capacity of foundations on the horizontal ground to calculate the damage loads. However, there is no complete solution for inclined loads for foundations on slopes [2].
The foundation bearing capacity and slope stability are both related to the limit state of the system and should be equivalent in terms of failure mechanisms [3]. Thus, the determination of the failure mechanism and the limit state is a challenging problem in the study of the ultimate inclined load of the foundation adjacent to the slope. The logarithmic spiral failure model was assumed using the limit equilibrium method [4]. The failure models based on the incremental displacement vectors were searched using FELA, e.g., the influence of inclined and eccentric loading on the bearing capacity of a strip footing placed on the reinforced cohesionless soil slope by using lower bound FELA [5]; the finite element program OptumG2 was used to study the undrained bearing capacity of an inclined loaded strip footing near a cohesive slope with a spatial variability of the undrained shear strength [6]. Fast Lagrangian analysis of the finite difference code continuum [7] was used to numerically analyze the bearing capacity of a strip footing near a cohesionless slope under a central inclined load. The determination of the failure model can be regarded as a nonlinear and nonsmooth global optimization problem. It is difficult to optimize the load problem with the existence of multiple local minima [8].
CMOC calculated the ultimate load starting from the stress state of the slope surface and assumed the outermost slip line as the critical slip surface [9][10][11]. According to the Mohr-Coulomb failure criterion, every slip line may be a slip surface. Only slip lines with a minimum safety factor are critical slip surfaces. The strength reduction method can directly obtain the failure modes, but the instability criteria need further study [12,13], e.g., there is no guidance on the selection of the convergence criteria or the optimal number of iterations, and the sharp point immediately is difficult to find when the displacement curve is relatively smooth. CMOC considered that the critical slope contour is concave [14][15][16], e.g., an instability criterion was proposed by [17] which is only applicable to the state where the critical slope contour intersects the slope toe. In this study, a convex critical slope contour was found and a failure mechanism was introduced to calculate the ultimate inclined load of the foundation adjacent to the slope. The influence of geometrical and mechanical parameters on the proposed method and the disadvantages of CMOC are studied.

Slip line equations
The equations of the slip line obtained by [14] are briefly introduced in this section. The relationship between stress components and principal stresses is as follows: where σ x , σ y , τ xy and τ yx represent the normal and shear stress in x and y directions, σ 1 and σ 3 are the maximum and minimum principal stresses, θ is the angle between σ 1 and the x-axis. The formula of characteristic stress σ is introduced using Mohr-Coulomb criterion: where c and φ are cohesion and internal friction angle. Substituting Eqs (4) and (5) into Eqs (1)-(3), and the expressions of the normal and shear stress are given as follows: The seismic differential equations are given as follows: where γ represents the unit weight. The limit equilibrium equations can be obtained by substituting Eqs (6)-(8) into Eqs (9) and (10): ð1 þ sinφcos2yÞ @s @x þ sinφsin2y @s @y À 2ssinφðsin2y @y @x À cos2y @y @y sinφsin2y @s @x þ ð1 À sinφcos2yÞ @s @y þ 2ssinφðcos2y @y @x þ sin2y @y @y Þ ¼ g ð12Þ Supplementary full differential equations: According to the definition of the characteristic lines, α and β families of the characteristic line equations can be obtained by solving the Eqs (11)- (14): ds À 2stanφdy ¼ gðdy À tanφdxÞ ð16Þ where m ¼ p 4 À φ 2 is the average angle between two families of slip line. The characteristic line Eqs (15)- (18) are approximately solved by the finite difference method: ðs À s a Þ À 2s a ðy À y a Þtanφ ¼ g½ðy À y a Þ À ðx À x a Þtanφ� ð20Þ The following Eqs (23)-(26) can be derived from Eqs (19)-(22). Note that the two equations in Eqs (24) or (26) yield the same result.

Boundary condition of inclined load
The closer the inclined load is to the slope crest, the more unstable the slope is. Thus, this paper only studies the case where the distance between the inclined load and the slope crest is assumed to be zero. As shown in Fig 1, three kinds of boundary value problems, e.g., Cauchy boundary (Active zone OAB), Degenerative Riemann boundary (Transition zone OBC), and Mixed boundary (Passive zone OCD), are needed to solve for calculation of the slip line field and the critical slope contour (i.e., line OD). Fig 2, θ 1 and σ 1 of the M α and M β points in the line OA can be derived using the Mohr-Coulomb failure criterion

Cauchy boundary. As shown in
Thus, the expression of θ 1 is: where δ is the inclination angle, q is the inclined load  imposed at the slope top surface. Thus, the expression of σ 1 is: 2.2.2 Degenerative Riemann boundary. The introduction of degenerative Riemann value is shown in S1 Appendix. The known σ 2 and θ 2 of point O in zone OBC are: where θ 3 can be calculated by the Eq (36) in Section 2.2.3, k = 0~n, n is the point partition of the Riemann boundary.

Mixed boundary.
The first known point M b of the critical slope contour is point O in the zone OCD. According to Eq (30), 3 can be obtained by substituting (30) into Eq (33): According to Eqs (35) and (36): According to Eq (30), σ 3 is a constant. According to Eq (32), σ 1 increases with q increasing. Thus, Δθ increases as σ 1 and q increase according to Eq (37). As shown in Fig 1(A) and 1(B), the critical slope contour OD changes from concave to convex with Δθ increasing.

Definition
For the convenience of calculation, the slope toe is defined as the coordinate origin. The position of the critical slope contour calculated from the method of characteristics varies with the increase of the inclined load q i = q 0 +i�Δq, where q 0 is the initial inclined load, Δq is the inclined load increment, i = 1, 2 ������n. As shown in Fig 3(A), the failure mechanism is proposed for calculating the ultimate inclined load q u : (1) the critical slope contour and the slope surface intersect at the slope crest (i.e., the first intersection) when q i <q u (stable state); (2) the critical slope contour and the slope surface intersect at the slope surface (i.e., the second intersection) when q i >q u (unstable state); (3) the critical slope contour is the critical boundary formed by transition point from one intersection to two intersections, i.e., q i is q u when the critical slope contour and the slope surface is at the critical position where two intersections will occur. The right-most slip line, i.e., the curve ABCD in Fig 1, is not the critical slip surface in this study. The instability criterion proposed by [17] is a special case of the proposed mechanism when the second intersection is the slope toe. The calculation flow chart is shown in Fig 3(B).

PLOS ONE
New failure mechanism for evaluating ultimate inclined load adjacent to slope

Verification
The parameters of the cases are γ = 20kN/m 3 , c = 20kPa, φ = 30 0 , η = 30 0 , slope height H = 2m, B = 2m, δ = 15 0 , i.e., the strength ratio c/γB = 0.5 and H/B = 1.0. The normalized ultimate inclined load factor is N u = q u /γB. The slip line fields calculated by the proposed method are shown in Fig 4. As expected, the intersection of the critical slope contour and the slope surface changes from one to two with N i = q i /γB increasing. The critical slope contour and the slope surface have one intersection (i.e., the slope crest (x, y) 1 = (3.46, 2.0)) when N i increases from N 1 = 2.575 to N 2 = 5.075 (as shown in Fig 4(A) and 4(B)), and those have two intersections when N i >5.075, e.g., (x, y) 2 = (1.51, 0.87) when N 3 = 6.075 (as shown in Fig 4(C)). According to Fig 3(A), N u = N 2 = 5.075 (i.e., q u = 203kPa). Fig 4(A)-4(C) show that Δθ increases with N i = q i /γB increasing, e.g., Δθ increases from 3.2 0 to 45.79 0 as N i increases from 2.575 to 6.075 (i.e., q i increases from 103kPa to 243 kPa). The critical slope contour changes from concave to convex as Δθ increases as shown in Fig 4(D).
In    When δ = 0 0 and φ = 40 0 , N u values are 15.8 (q u = 632kPa) and 24.3 (q u = 972kPa) calculated by the proposed method and CMOC. As shown in Fig 6(B)-6(E), The safety factor (FSs) are calculated using SLIED5.0 and FLAC7.0 when N u = 15.8 (q u = 632kPa) and N u = 24.3 (q u = 972kPa) are imposed at the slope top surface. FS is equal to 1.0 when the ultimate inclined load is imposed on the slope top surface. Compared with CMOC (i.e., FS = 0.922 and 0.86), FS = 1.035 and 1.02 calculated with the proposed method are more closed to 1.0. The reason is that the failure mode becomes shallower with φ increasing [18], e.g., φ increases from 20 0 to 40 0 in this case. CMOC assumed that the outermost slip line is the critical slip surface to obtain larger failure modes and overestimated q u .

η and c/γB variation
When H/B = 2.5, c/γB = 0.25, φ = 30 0 , η varies from 15 0 to 60 0 , and tan(δ) varies from 0 to 0.5, comparison of N u values calculated by the proposed method and FELA are shown in Fig 7(A), where UB and LB-FELA solutions are calculated using OptumG2 [19]. For η = 15 0 , 30 0 , and 60 0 , N u values calculated by the proposed method are almost between those of UB and LB-FELA, except in the cases of tan(δ) = 0.3 and η = 60 0 . When tan(δ) = 0.3 and η = 60 0 , N u = 1.1 and 1.06 are calculated by the proposed method and the UB-FELA, and the error is 3.6%. For η = 45 0 , the result of the proposed method is slightly larger than those of the UB-FELA and the maximum error is 7.4%, e.g., N u = 3.2 and 3.0 are calculated by the proposed method, and the UB-FELA when tan(δ) is 0. Fig 7(B) shows that N u calculated by CMOC (Fig 11(b) in [10]) are much larger than those of the proposed method. Thus, CMOC overestimated N u , e.g., N u values obtained by CMOC are 3.4, 3.4, 3.4, and 2.1 for η = 15 0 , 30, 45 0 , and 60 0 when tan(δ) is 0.5. Under the same condition, N u values obtained by the proposed method are 1.7, 1.3, 0.9, and 0.7. The proposed method is more reasonable since the N u values of the proposed method are close to those of the FELA as shown in Fig 7(A). Table 1 shows that the FS related to the N u of the proposed method, calculated by SLIED5.0, is close to 1.0.

Discussion
This section discusses in detail the reason why the current method misjudges the ultimate inclination load when c/γB = 0.25 in  [10]) was assumed by CMOC. Fig 9(B)-9(E) show that the failure models were obtained using SLIDE5.0 and FLAC7.0 when q u = 101kPa, 306kPa are imposed at the slope top surface.

Conclusion
1. The boundary value problems were derived for calculating the slip lines and the critical slope contour when the inclined load is imposed at the slope top surface. The angle (Δθ) between the maximum principal stress and the x-axis in the Degenerative Riemann boundary (Transition zone) increases as the inclined load (q) imposed on the slope top surface increases. The critical slope contour changes from concave to convex when Δθ and q increase.
2. The critical slope contour shifts from the inside of the slope to the outside of the slope as q increases. The critical slope contour and slope surface have one intersection when the critical slope contour is inside the slope (stable state), and they have two intersections when the critical slope contour is outside the slope (unstable state). The critical slope contour and the slope surface are at the critical position where two intersections will occur, and q is the ultimate inclined load q u .
3. When the strength ratio is large (e.g., c/γB = 0.5), q u and the failure model calculated by the proposed method are consistent with those of CMOC and LB-FELA. According to the  definition of the ultimate inclined load (i.e., that the safety factor is equal to 1.0 when the ultimate inclined load is imposed on the slope top surface), the proposed method is more reasonable since the safety factor calculated by the proposed method is close to 1.0. The proposed method is close to the FELA when the strength ratio is small (e.g., c/γB = 0.25).

PLOS ONE
4. CMOC overestimated q u or gave an unreasonable result when the friction angle is large (e.g., φ = 40 0 ) and the strength ratio is small (e.g., c/γB = 0.25). The reason is that CMOC assumed the outermost slip line as the critical slip surface to obtain a larger failure model since the failure model becomes shallower with the friction angle (φ) increasing and the strength ratio decreasing. 5. CMOC considers that the critical slope contour is concave. This study finds that the critical slope contour is convex when the inclined load on the top of the slope increases. The proposed method does not need to assume or search the failure models, and it can solve the difficult problem that the failure models are not easy to determine. The proposed method gives the criterion of slope in the limit state without the strength parameters reduction, and it is simple and robust.